There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong
(L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$

Can someone provide an example?

I have found the following thread. But there, the underlying structure is a ring. Hence, an answer to the current question is automatically an answer to the old question as well. EDIT: Sorry, the other question asks for finite commutative ring, and this won't be possible in the case of fields…

@Sanchez This can be a good example as one knows that $\mathbb Z[\sqrt 2]$ and $\mathbb Z[\sqrt 3]$ are the rings of algebraic integers of $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 3)$, respectively and both are euclidean. But their groups of units are not so easy to handle. (Anyway, they are both isomorphic to $\{\pm 1\}\times\mathbb Z$ if I'm not wrong.) That's why I've preferred imaginary quadratic number fields.
–
user26857Jan 1 '14 at 22:00